Thursday:  October 26th, 2000 Mathematics Colloquium: Alex Postnikov, University of California at Berkeley, will speak on "Schubert Calculus and Quantum Cohomology", at 4:00PM in Math 501. Refreshments at 3:30PM in Math 402. Abstract: A long standing open problem in Schubert calculus is to give a combinatorial interpretation to the intersection numbers of Schubert varieties in flag manifolds. Equivalently, these numbers are the structure constants of the cohomology ring of the flag manifold. They generalize the famous Littlewood-Richardson coefficients. I will discuss the generalization of this problem to the quantum cohomology ring of the flag manifold, which is a multiparameter deformation of the usual cohomology ring. Its structure constants are called the Gromov-Witten invariants. Many results from the Schubert calculus, such as Monk's and Pieri's rules, naturally extend to the quantum cohomology. Remarkably, the Gromov-Witten invariants seem to be more symmetric objects than the Schubert intersection numbers. They possess new properties that could not be easily detected on the "classical'' level of the Schubert intersection numbers.